The Stokes problem of fluid mechanics, Riesz transform, and the Helmholtz-Hodge decomposition : probabilistic methods and their representations

Abstract:

The flow of incompressible, viscous fluids in R³ is governed by the non-linear Navier-Stokes equations. Two common linearizations of the Navier-Stokes equations, the Stokes equations and the Oseen equations, are studied in this thesis using probabilistic methods.
The incompressibility condition presents new challenges for the well known theory relating partial differential equations and stochastic processes. In this thesis we construct probabilistic solutions to the incompressible Stokes equations in the absence of boundaries, in the case of the half space, and we make some observations for general domains.
Also we give probabilistic representations for the iterated Riesz transforms, the Helmholtz-Hodge decomposition on domains with smooth boundaries as well as the free space, and the solutions to the Neumann problems on exterior domains and the half space.